After applying L'Hospital's rule 3 times I could just see it getting untidy and I couldn't see my and mistakes in my working so I checked against WolframAlpha which showed that L'Hospital's rule is needed 10 times to get to the solution which seems like a lot of work for a small question. Is there a more efficient method for solving this limit that I'm unaware of?

EDIT: Sorry if this should've been in the pre calculus section, I wasn't sure.

if you're taking the limit as it goes to infinity you can drop any constants and only keep the highest power of x about since as you go to infinity those are going to be insignificant
doing this to your equation gives
[tex]lim \frac{(-2)^4x^{12}}{x^{12}}=lim 16 = 16[/tex]